Metamath Proof Explorer

Theorem mdandyvr12

Description: Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly. (Contributed by Jarvin Udandy, 7-Sep-2016)

Ref Expression
Hypotheses mdandyvr12.1 ⊢ φ ↔ ζ
mdandyvr12.2 ⊢ ψ ↔ σ
mdandyvr12.3 ⊢ χ ↔ φ
mdandyvr12.4 ⊢ θ ↔ φ
mdandyvr12.5 ⊢ τ ↔ ψ
mdandyvr12.6 ⊢ η ↔ ψ
Assertion mdandyvr12 ⊢ χ ↔ ζ ∧ θ ↔ ζ ∧ τ ↔ σ ∧ η ↔ σ

Proof

Step Hyp Ref Expression
1 mdandyvr12.1 ⊢ φ ↔ ζ
2 mdandyvr12.2 ⊢ ψ ↔ σ
3 mdandyvr12.3 ⊢ χ ↔ φ
4 mdandyvr12.4 ⊢ θ ↔ φ
5 mdandyvr12.5 ⊢ τ ↔ ψ
6 mdandyvr12.6 ⊢ η ↔ ψ
7 2 1 3 4 5 6 mdandyvr3 ⊢ χ ↔ ζ ∧ θ ↔ ζ ∧ τ ↔ σ ∧ η ↔ σ